How do you add or subtract sinusoidal AC quantities using phasors and complex numbers?

In one line

To add or subtract sinusoids of the same frequency, write each as a phasor in rectangular form \( (a + jb) \), add or subtract the real and imaginary parts separately, then convert back to magnitude and angle.

Explained simply

Two people pull a box with ropes at different angles. You cannot just add the two pulls as numbers; you add them as arrows (vectors). AC voltages that are out of phase behave the same way: 3 V and 4 V at 90° to each other give 5 V, not 7 V.

Technically, a phasor \( A\angle\theta \) has rectangular form \( A\cos\theta + jA\sin\theta \), where \( j = \sqrt{-1} \) rotates by 90°. Addition and subtraction are done in rectangular form; multiplication and division are easier in polar form.

How it works

1. Write each sinusoid as a phasor \( A\angle\theta \) (using peak values or RMS values, but the same for all).
2. Convert to rectangular: \( x = A\cos\theta \), \( y = A\sin\theta \).
3. Add or subtract real parts and imaginary parts separately.
4. Convert back: magnitude \( = \sqrt{x^2+y^2} \), angle \( = \tan^{-1}(y/x) \) (check the quadrant).
5. Write the result as a time function with the same \( \omega \).

Multiplication: multiply magnitudes, add angles. Division: divide magnitudes, subtract angles.

Key formula

\( A\angle\theta = A\cos\theta + jA\sin\theta \); \( (x_1 + jy_1) \pm (x_2 + jy_2) = (x_1 \pm x_2) + j(y_1 \pm y_2) \)

\( A \) = magnitude (same unit as the quantity, V or A), \( \theta \) = phase angle, \( x \) = real (horizontal) component, \( y \) = imaginary (vertical) component, \( j \) = 90° operator.

Quick example

\( i_1 = 10\sin\omega t \) and \( i_2 = 20\sin(\omega t + 60^\circ) \) A. In rectangular form: \( 10 + j0 \) and \( 10 + j17.32 \). Sum \( = 20 + j17.32 \). Magnitude \( = \sqrt{400 + 300} = 26.46 \) A; angle \( = \tan^{-1}(17.32/20) = 40.9^\circ \). So \( i = 26.46\sin(\omega t + 40.9^\circ) \) A. Simple addition would have wrongly given 30 A.

Common mistakes

• Adding magnitudes directly when the phase angles differ.
• Taking the wrong quadrant for \( \tan^{-1} \) when the real part is negative.
• Mixing peak and RMS values in the same sum.

How MU asks it

A common 5-mark numerical: "Find the resultant of \( v_1 = \dots \) and \( v_2 = \dots \) and express it in the form \( V_m\sin(\omega t \pm \phi) \)", sometimes with subtraction. For full marks: show the polar-to-rectangular conversion of each, the sum, conversion back to polar, the final time expression, and a neat phasor diagram.